3.628 \(\int \frac{\left (a+b x^4\right )^2}{x^4} \, dx\)

Optimal. Leaf size=26 \[ -\frac{a^2}{3 x^3}+2 a b x+\frac{b^2 x^5}{5} \]

[Out]

-a^2/(3*x^3) + 2*a*b*x + (b^2*x^5)/5

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Rubi [A]  time = 0.0280088, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{a^2}{3 x^3}+2 a b x+\frac{b^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^2/x^4,x]

[Out]

-a^2/(3*x^3) + 2*a*b*x + (b^2*x^5)/5

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Rubi in Sympy [A]  time = 5.01069, size = 22, normalized size = 0.85 \[ - \frac{a^{2}}{3 x^{3}} + 2 a b x + \frac{b^{2} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**2/x**4,x)

[Out]

-a**2/(3*x**3) + 2*a*b*x + b**2*x**5/5

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Mathematica [A]  time = 0.00121466, size = 26, normalized size = 1. \[ -\frac{a^2}{3 x^3}+2 a b x+\frac{b^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^2/x^4,x]

[Out]

-a^2/(3*x^3) + 2*a*b*x + (b^2*x^5)/5

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Maple [A]  time = 0.005, size = 23, normalized size = 0.9 \[ -{\frac{{a}^{2}}{3\,{x}^{3}}}+2\,abx+{\frac{{b}^{2}{x}^{5}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^2/x^4,x)

[Out]

-1/3*a^2/x^3+2*a*b*x+1/5*b^2*x^5

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Maxima [A]  time = 1.49092, size = 30, normalized size = 1.15 \[ \frac{1}{5} \, b^{2} x^{5} + 2 \, a b x - \frac{a^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^2/x^4,x, algorithm="maxima")

[Out]

1/5*b^2*x^5 + 2*a*b*x - 1/3*a^2/x^3

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Fricas [A]  time = 0.216637, size = 35, normalized size = 1.35 \[ \frac{3 \, b^{2} x^{8} + 30 \, a b x^{4} - 5 \, a^{2}}{15 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^2/x^4,x, algorithm="fricas")

[Out]

1/15*(3*b^2*x^8 + 30*a*b*x^4 - 5*a^2)/x^3

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Sympy [A]  time = 1.04259, size = 22, normalized size = 0.85 \[ - \frac{a^{2}}{3 x^{3}} + 2 a b x + \frac{b^{2} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**2/x**4,x)

[Out]

-a**2/(3*x**3) + 2*a*b*x + b**2*x**5/5

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GIAC/XCAS [A]  time = 0.222841, size = 30, normalized size = 1.15 \[ \frac{1}{5} \, b^{2} x^{5} + 2 \, a b x - \frac{a^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^2/x^4,x, algorithm="giac")

[Out]

1/5*b^2*x^5 + 2*a*b*x - 1/3*a^2/x^3